U.S. patent application number 13/586712 was filed with the patent office on 2013-08-15 for method to optimize perforations for hydraulic fracturing in anisotropic earth formations.
The applicant listed for this patent is Florian Karpfinger, Brice Lecampion, Romain Charles Andre Prioul, George Waters. Invention is credited to Florian Karpfinger, Brice Lecampion, Romain Charles Andre Prioul, George Waters.
Application Number | 20130206475 13/586712 |
Document ID | / |
Family ID | 48944682 |
Filed Date | 2013-08-15 |
United States Patent
Application |
20130206475 |
Kind Code |
A1 |
Prioul; Romain Charles Andre ;
et al. |
August 15, 2013 |
METHOD TO OPTIMIZE PERFORATIONS FOR HYDRAULIC FRACTURING IN
ANISOTROPIC EARTH FORMATIONS
Abstract
The subject disclosure relates to determining an optimum
orientation for perforations around the circumference of a
subsurface borehole and optimum wellbore fluid initiation pressure
for hydraulic fracturing in anisotropic formations.
Inventors: |
Prioul; Romain Charles Andre;
(Somerville, MA) ; Karpfinger; Florian;
(Stavanger, NO) ; Waters; George; (Oklahoma City,
OK) ; Lecampion; Brice; (Cambridge, MA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Prioul; Romain Charles Andre
Karpfinger; Florian
Waters; George
Lecampion; Brice |
Somerville
Stavanger
Oklahoma City
Cambridge |
MA
OK
MA |
US
NO
US
US |
|
|
Family ID: |
48944682 |
Appl. No.: |
13/586712 |
Filed: |
August 15, 2012 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
61524042 |
Aug 16, 2011 |
|
|
|
Current U.S.
Class: |
175/2 ;
166/250.1 |
Current CPC
Class: |
G01V 1/46 20130101; E21B
43/117 20130101; G01V 1/50 20130101; E21B 43/263 20130101; E21B
43/119 20130101; E21B 49/00 20130101; E21B 43/26 20130101 |
Class at
Publication: |
175/2 ;
166/250.1 |
International
Class: |
E21B 43/26 20060101
E21B043/26; E21B 43/263 20060101 E21B043/263 |
Claims
1. A method for determining a perforation orientation for hydraulic
fracturing in an anisotropic earth formation comprising:
determining anisotropic rock properties; determining far-field
stresses in the anisotropic earth formation; determining borehole
stresses in the anisotropic earth formation; and determining an
optimum perforation orientation and optimum wellbore fluid
initiation pressure.
2. The method according to claim 1 further comprising perforating a
well in the determined optimum perforation orientation.
3. The method according to claim 1 wherein the determining
anisotropic rock properties further comprises: acquisition of a
sonic log with a 3D deviation survey; data processing to
characterize borehole sonic anisotropy.
4. The method according to claim 3 wherein the acquisition of the
sonic log uses a monopole mode.
5. The method according to claim 3 wherein the acquisition of the
sonic log uses a dipole mode.
6. The method according to claim 3 wherein the acquisition of the
sonic log uses a monopole mode, a dipole mode or a Stoneley mode,
or any combination of.
7. The method according to claim 2 wherein the well is a deviated
well.
8. The method according to claim 2 wherein the well is a horizontal
well.
9. The method according to claim 2 wherein the well is a vertical
well.
10. The method according to claim 2 wherein the perforating the
well in the optimum perforation orientation is performed with at a
shaped charge.
11. A method for perforating a well traversing a subterranean area
including one or more transversely isotropic formations with a
tilted axis of symmetry comprising: determining formation
properties; determining far-field stresses in the formation;
determining borehole stresses in the formation; determining an
optimum perforation orientation and optimum wellbore fluid
initiation pressure; and perforating the well in the determined
optimum perforation orientation.
12. The method according to claim 11 wherein the well comprises one
or more portions of a group consisting of a deviated portion, a
horizontal portion, or a vertical portion.
13. The method according to claim 11 wherein the perforating of the
well is done with one or more shaped charges.
14. The method according to claim 11 further comprising:
determination of borehole stresses in the well at different depths;
positioning perforation clusters at one or more depth points with
borehole stresses similar to a previous perforation depth; and
perforating the well at the one or more depth points for placement
of hydraulic fracturing stages along the well.
15. A method for hydraulic fracturing in an anisotropic earth
formation comprising: determining anisotropic rock properties;
determining far-field stresses in the anisotropic earth formation;
determining borehole stresses in the anisotropic earth formation;
determining an optimum perforation orientation and optimum wellbore
fluid initiation pressure; perforating a well in the determined
optimum perforation orientation; and hydraulic fracturing the well
at a pressure at least at the optimum wellbore fluid initiation
pressure.
16. The method according to claim 15 wherein the well comprises one
or more portions of a group consisting of a deviated portion, a
horizontal portion, or a vertical portion.
17. The method of claim 15 in which the determining anisotropic
rock properties comprises acquisition of wireline sonic logs with
all modes with a 3D deviation survey.
18. The method of claim 15 in which the determining anisotropic
rock properties comprises acquisition of logging while drilling
sonic logs with all modes with a 3D deviation survey.
19. The method of claim 15 in which the perforating of the well
comprises one or more shape charges.
20. The method according to claim 17 wherein the acquisition of the
sonic log uses a monopole mode, a dipole mode or a Stoneley mode,
or any combination of.
Description
RELATED APPLICATIONS
[0001] This application claims the benefit of a related U.S.
Provisional Application Ser. No. 61/524,042 filed Aug. 16, 2011,
entitled "METHOD TO OPTIMIZE PERFORATIONS FOR HYDRAULIC FRACTURING
IN ANISOTROPIC EARTH FORMATIONS," to Romain Prioul et al., the
disclosure of which is incorporated by reference herein in its
entirety.
FIELD
[0002] The subject disclosure generally relates to the field of
geosciences. More particularly, the subject disclosure relates to
the determination of the orientation around the circumference of a
subsurface borehole and the wellbore fluid initiation pressure that
is optimum for perforation operations for hydraulic fracturing in
anisotropic formations.
BACKGROUND
[0003] Perforation techniques are widely used in the oil and gas
industry both for enhancing hydrocarbon production by minimizing
sand production and for hydraulic fracture stimulation initiation.
Citing a comprehensive review on the topic, "the process of
optimizing stimulation treatments uses orientated perforations to
increase the efficiency of pumping operations, reduce treatment
failures and improve fracture effectiveness. Completion engineers
develop oriented-perforating strategies that prevent sand
production and enhance well productivity by perforating to
intersect natural fractures or penetrate sectors of a borehole with
minimal formation damage." See Almaguer et al., "Orienting
perforations in the right direction", Oilfield Review, Volume 1,
Issue 1, Mar. 1, 2002.
[0004] Hydraulic fractures initiate and propagate from positions
around the circumference of the open borehole wall that offer the
least resistance in terms of stress and rock strength conditions.
If the formation material properties (e.g. elastic stiffness and
strength) are isotropic and homogeneous and if the material is
intact (free of natural fractures or flaws), it is generally
accepted that the fracture initiation occurs at the locus around
the borehole where the tensile stress is maximum. The stress
conditions at the borehole wall in such formation depends on the
local stress orientations and magnitudes (local principal stress
tensor), the orientation of the borehole and a material property
called Poisson's ratio (if the formation is assumed elastic).
[0005] One definition of an optimum perforation orientation is the
orientation around the circumference of a subsurface borehole wall
and the wellbore fluid initiation pressure that corresponds to the
minimum principal stress at the borehole wall (rock mechanics
convention is chosen here with positive compressive stress)
reaching the tensile strength of the rock. Consequently, the
optimum perforation orientation will ultimately lower the treatment
pressure during hydraulic fracturing therefore lowering the energy
requirement of a job. It will also result in a "smoother" fracture
near the wellbore (i.e. less near wellbore tortuosity) in which
proppant can be placed more effectively.
[0006] Perforation orientations may be designed with the following
typical steps:
1. A rock property called Poisson's ratio v is estimated along the
well most commonly using compressional V.sub.p and shear V.sub.s
sonic log data from formula
v=0.5(V.sub.p.sup.2-2V.sub.s.sup.2)/(V.sub.p.sup.2-V.sub.s.sup.2).
Other methods may also be used as is known in the art. 2. The
far-field stress field (or tensor), .sigma., and pore pressure,
P.sub.p, are characterized using direct or indirect stress
measurements, leading to three principal stress directions and
magnitudes (.sigma..sub.1>.sigma..sub.2>.sigma..sub.3) in the
subsurface. When one principal stress is vertical and called
.sigma..sub.V, the following convention is used .sigma..sub.H, and
.sigma..sub.h for, respectively, the maximum and minimum horizontal
principal stresses. For a recent review of the existing methods,
see Hudson, J. A., F. H. Cornet, R. Christiansson, "ISRM Suggested
Methods for rock stress estimation Part 1: Strategy for rock stress
estimation", International Journal of Rock Mechanics & Mining
Sciences 40 (2003) 991998; Sjoberg, J., R. Christiansson, J. A.
Hudson, "ISRM Suggested Methods for rock stress estimation Part 2:
Overcoring methods", International Journal of Rock Mechanics &
Mining Sciences 40 (2003) 9991010; Haimson, B. C., F. H. Cornet,
"ISRM Suggested Methods for rock stress estimation Part 3:
hydraulic fracturing (HF) and/or hydraulic testing of pre-existing
fractures (HTPF)", International Journal of Rock and U.S. Pat. No.
8,117,014 to Prioul et al., entitled "Methods to estimate
subsurface deviatoric stress characteristics from borehole sonic
log anisotropy directions and image log failure directions". 3.
Given known well orientation as a function of depth, defined by two
angles (well azimuth and deviation), the principal stress tensor
.sigma.=[(.sigma..sub.1 0 0; 0 .sigma..sub.2 0; 0 0 .sigma..sub.3]
can be transformed using tensor rotation into a wellbore frame for
example using so-called TOH-frame stress tensor
.sigma..sub.TOH=[.sigma..sub.xx.sup.TOH
.sigma..sub.xy.sup.TOH.sigma..sub.xz.sup.TOH .sigma..sub.xy.sup.TOH
.sigma..sub.yy.sup.TOH .sigma..sub.yz.sup.TOH;
.sigma..sub.xz.sup.TOH .sigma..sub.yz.sup.TOH
.sigma..sub.zz.sup.TOH]. The TOH (top of the hole) frame is a
coordinate system tied to the tool/borehole. Hence, its x- and
y-axes are contained in the plane perpendicular to the
tool/borehole, and the z-axis is pointing along the borehole in the
direction of increasing depth. The x-axis of the TOH frame is
pointing to the top of the borehole, the y-axis is found by
rotating the x-axis 90 degrees in the tool plane in a direction
dictated by the right hand rule (thumb pointing in the positive
z-direction). Given a known internal wellbore pressure, P.sub.w,
borehole stresses (or near-field) are then computed using
well-known Kirsch analytical expressions, (See Ernst Gustav Kirsch.
Die Theorie der Elastizitat and die Bedurfnisse der
Festigkeitslehre. "Zeitschrift des Vereines Deutscher Ingenieure",
42(29):797-807, 1898; Y. Hiramatsu and Y. Oka. "Stress around a
shaft or level excavated in ground with a three-dimensional stress
state"; Kyoto Teikoku Daigaku Koka Daigaku kiyo, page 56, 1962; Y.
Hiramatsu and Y. Oka. "Determination of the stress in rock
unaffected by boreholes or drifts, from measured strains or
deformations", International Journal of Rock Mechanics and Mining
Sciences & Geomechanics Abstracts, volume 5, pages 337-353.
Elsevier, 1968), for the total stresses at the borehole wall for an
arbitrary orientation of the borehole relative to the far-field
in-situ stress tensor, as follows in cylindrical coordinates:
.sigma..sub.rr=P.sub.w,
.sigma..sub..theta..theta.=.sigma..sub.xx.sup.TOH+.sigma..sub.yy.sup.TOH-
-2(.sigma..sub.xx.sup.TOH-.sigma..sub.yy.sup.TOH)cos
2.theta.-4.sigma..sub.xy.sup.TOH sin 2.theta.-P.sub.w,
.sigma..sub.zz.sup.TOH=.sigma..sub.zz.sup.TOH-2v(.sigma..sub.xx.sup.TOH--
.sigma..sub.yy.sup.TOH)cos 2.theta.-4v.sigma..sub.xy.sup.TOH sin
2.theta.,
.sigma..sub..theta.z=2(.sigma..sub.yz.sup.TOH cos
.theta.-.sigma..sub.xz.sup.TOH sin .theta.),
.sigma..sub.r.theta.=.sigma..sub.rz=0,
where v is the Poisson's ratio, .theta. is the azimuthal angle
around the borehole circumference measured clockwise from a
reference axis (e.g. TOH). Equations to compute borehole stresses
away from the borehole wall at a desired radial position into the
formation are also available. 4. Then, the ideal perforation
orientation for tensile initiation is found for the azimuthal
position .theta..sub.t and the wellbore fluid initiation pressure
P.sub.w.sup.init where the minimum principal stress at the borehole
wall is given by
.sigma. t = .sigma. zz + .sigma. .theta..theta. 2 - ( .sigma. zz -
.sigma. .theta..theta. 2 ) 2 + .sigma. .theta. z 2 = - To + Pp ,
##EQU00001##
where To is the tensile strength of the rock and Pp is the pore
pressure. 5. Once the optimum orientation is known a perforation
tool is lowered in the well. The perforation tool perforates the
well in an optimum orientation obtained from the previous step.
[0007] For a stress field with one principal stress that is
vertical (.sigma..sub.V), we consider the special cases of well
orientations where the azimuthal position .theta..sub.t is always
in a principal direction:
(a) For vertical wells, the azimuthal position .theta..sub.t is the
minimum hoop stress (minimum of .sigma..sub..theta..theta.) which
is always in the direction of the maximum horizontal principal
stress, .sigma..sub.H. (b) For horizontal wells drilled in the
direction of a principal stress direction (.sigma..sub.H or
.sigma..sub.h), the azimuthal position .theta..sub.t is also the
one given by the minimum hoop stress (minimum of
.sigma..sub..theta..theta.), i.e. is pointing to the top of the
hole if .sigma..sub.V is greater than the horizontal stress
orthogonal to the borehole, or to the side of the hole if
.sigma..sub.V is smaller than the horizontal stress orthogonal to
the borehole.
[0008] If the well is deviated, in such a stress field the
orientation is not aligned with a principal stress direction and
there is no obvious solution for .theta..sub.t as it also depends
on the wellbore fluid initiation pressure so the orientation is
computed numerically. See Peska, P. & Zoback, M., Compressive
and tensile failure of inclined well bores and determination of in
situ stress and rock strength, Journal of Geophysical Research,
1995, 100, 12,791-12,811.
[0009] When the earth formation has material properties that are
directions dependent, i.e. anisotropic, steps 1, 2 and 3 above are
not valid anymore and depend on the anisotropy of the rock.
Although some studies have been completed on the impact of the
anisotropy on the borehole stress concentration (i.e. step 3),
those studies have focused on the wellbore stability issues and mud
weight requirements to prevent wellbore collapse (shear) and
tensile fracturing (tensile), and not on a workflow to assess the
best perforation orientation.
SUMMARY
[0010] This summary is provided to introduce a selection of
concepts that are further described below in the detailed
description. This summary is not intended to identify key or
essential features of the claimed subject matter, nor is it
intended to be used as an aid in limiting the scope of the claimed
subject matter.
[0011] In embodiments of the subject disclosure methods are
disclosed to determine the optimum orientation for perforations
around a circumference of a subsurface borehole and the wellbore
fluid initiation pressure that is for hydraulic fracturing in
anisotropic formations.
[0012] In embodiments of the subject disclosure methods are
disclosed for determining a perforation orientation for hydraulic
fracturing in an anisotropic earth formation. In embodiments the
method comprises the steps of determining anisotropic rock
properties; determining far-field stresses in the anisotropic earth
formation; determining borehole stresses in the anisotropic earth
formation; determining an optimum perforation orientation; and
perforating a well in the determined optimum perforation
orientation.
[0013] Further features and advantages of the subject disclosure
will become more readily apparent from the following detailed
description when taken in conjunction with the accompanying
drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
[0014] The subject disclosure is further described in the detailed
description which follows, in reference to the noted plurality of
drawings by way of non-limiting examples of embodiments of the
subject disclosure, in which like reference numerals represent
similar parts throughout the several views of the drawings, and
wherein:
[0015] FIG. 1 illustrates a schematic of the geographic and
borehole reference frames and the principal stress directions;
[0016] FIG. 2 illustrates the material coordinate system for a
transverse isotropic medium with a tilted symmetry axis (TTI);
[0017] FIG. 3 illustrates a workflow for determining perforation
orientations for hydraulic fracturing in anisotropic earth
formations; and
[0018] FIG. 4 illustrates an example of a perforation orientation
angle around a borehole.
DETAILED DESCRIPTION
[0019] The particulars shown herein are by way of example and for
purposes of illustrative discussion of the embodiments of the
subject disclosure only and are presented in the cause of providing
what is believed to be the most useful and readily understood
description of the principles and conceptual aspects of the subject
disclosure. In this regard, no attempt is made to show structural
details in more detail than is necessary for the fundamental
understanding of the subject disclosure, the description taken with
the drawings making apparent to those skilled in the art how the
several forms of the subject disclosure may be embodied in
practice. Furthermore, like reference numbers and designations in
the various drawings indicate like elements.
[0020] Embodiments of the subject disclosure relate to the
determination of the orientation around the circumference of a
subsurface borehole and the wellbore fluid initiation pressure that
is optimum for perforation operations for hydraulic fracturing in
anisotropic formations. In one non-limiting example, perforation
operations include shaped charge perforation operations.
[0021] Embodiments of the subject disclosure comprise methods which
are applicable to arbitrary well orientation, arbitrary stress
field and arbitrary elastic anisotropy of a formation.
[0022] Embodiments of the subject disclosure disclose a workflow
method comprising a plurality of steps for determining perforation
orientations for hydraulic fracturing in anisotropic earth
formations. The plurality of steps include determination of
anisotropic rock properties, determination of far-field stresses in
anisotropic formations, determination of borehole stresses in
anisotropic formations, determination of the optimum fracture
orientation and optimum initiation pressure, lowering in the well a
perforation tool and perforating the well in the direction of the
optimum orientation obtained from the previous steps. Anisotropic
rock properties and far-field stress properties may vary along the
well and borehole stresses may vary along the borehole, therefore,
the step of determining the borehole stresses in anisotropic
formations may be used to select the depth points of where to place
perforation clusters for a given hydraulic fracturing stage in rock
with similar near-wellbore stresses or similar wellbore fluid
initiation pressure. Therefore, the step of determining the
borehole stresses in anisotropic formations and the borehole
stresses may be used to determine how to place hydraulic fracturing
stages along the well.
[0023] The subject disclosure will be described in greater detail
as follows. First, a number of definitions useful to understanding
the subject disclosure are presented.
DEFINITIONS
Geometry and Coordinate Systems:
[0024] In the far-field an in-situ stress field is applied where
the principal stress tensor takes the form:
.sigma. = ( .sigma. H 0 0 0 .sigma. h 0 0 0 .sigma. v )
##EQU00002##
where .sigma..sub.H and .sigma..sub.h are the maximum and minimum
horizontal stresses, respectively, and .sigma..sub.v is the
vertical stress. FIG. 1 illustrates a schematic of the geographic
and borehole reference frames and the principal stress directions.
The geographic reference frame is the north-east-vertical (NEV)
frame whose x-axis points to the north, y-axis points to the east,
and z-axis points downward in vertical direction. The borehole
frame is the top-of-hole (TOH) frame whose z-axis points along the
borehole in the direction of increasing depth. The x-axis is in the
cross-sectional plane and points to the most upward direction, and
the y-axis is found by rotating the x-axis 90.degree. in the
cross-sectional plane in a direction dictated by the right-hand
rule. The orientation of the borehole is defined by the deviation
angle .alpha..sub.D and the azimuth angle .alpha..sub.A.
[0025] For the sake of simplicity, but without loss of generality,
we assume that the vertical stress .sigma..sub.v is always aligned
with the vertical component (V) of the NEV (north-east-vertical)
coordinates system. The horizontal stress field can be rotated by
an angle .gamma. measured between N (north) and .sigma..sub.H
towards E (east). For the computation of the borehole stress
concentration it is convenient to rotate the stress field in the
NEV frame into the top-of-hole borehole coordinate system,
hereafter called TOH (see definition above), i.e.
.sigma..sub.TOH=[.sigma..sub.xx.sup.TOH .sigma..sub.xy.sup.TOH
.sigma..sub.xz.sup.TOH .sigma..sub.xy.sup.TOH
.sigma..sub.yy.sup.TOH .sigma..sub.yz.sup.TOH;
.sigma..sub.xz.sup.TOH .sigma..sub.yz.sup.TOH
.sigma..sub.zz.sup.TOH]. The orientation of the borehole is defined
by the deviation angle .alpha..sub.D and the azimuth angle
.alpha..sub.A.
Elasticity Equation
[0026] The strain components .epsilon..sub.ij are related to the
stress components .sigma..sub.kl via the constitutive relation:
.epsilon..sub.ij=S.sub.ijkl.sigma..sub.kl
where S.sub.ijki is the fourth rank compliance tensor (and as
s.sub.ij if the 6.times.6 matrix contracted Voigt notation is
used). The inverse of the compliance tensor is the fourth rank
stiffness tensor defined as C.sub.ijki (and c.sub.ij in Voigt
notation). Rotation of the compliance tensor into the TOH frame can
be done by applying two Bond transformations to the 6.times.6 Voigt
notation compliance matrix s.sub.ij giving a new matrix noted
.sigma..sub.ij.
Material Anisotropy
[0027] Embodiments of the subject disclosure use an anisotropic
medium that is transversely isotropic rocks with a titled axis of
symmetry (called hereafter TTI). In general, this is the most
typical type of anisotropy encountered in the Earth, although it
should be understood that methods of the subject disclosure are not
restricted to TI media. The TTI medium is described by five elastic
constants in different notations as
(a) Elasticity notation: c.sub.11, c.sub.33, c.sub.13, c.sub.44 and
c.sub.66 are the five stiffness coefficients in Voigt notation of
the stiffness tensor entering in the elasticity relationship
between stress and strain. The orientation of the TI plane is
defined by two angles as depicted in FIG. 2, the dip azimuth
.beta..sub.A and the dip angle .beta..sub.D. FIG. 2 depicts the
material coordinate system for transverse isotropic medium with
tilted symmetry axis (called TTI) where .beta..sub.D is the dip of
the transverse isotropy plane and .beta..sub.A is the dip azimuth.
(b) Engineering notation: E.sub.v, and E.sub.h are the vertical and
horizontal Young's moduli (with respect to TI plane), v.sub.v and
v.sub.h the vertical and horizontal Poisson's ratios, and G.sub.v
the vertical shear moduli. The orientation of the TI plane is also
defined by two angles as depicted in FIG. 2, the dip azimuth and
the dip angle. (c) Geophysics notation: Vp0 and Vs0 are
respectively the compressional and shear velocities along the
symmetry axis and .epsilon., .delta., .gamma. are three
dimensionless parameters (called Thomsen parameters) and .rho. is
the rock bulk density. The orientation of the TI plane is also
defined by two angles as depicted in FIG. 2, the dip azimuth and
the dip angle.
Initiation Pressure
[0028] In the present disclosure, the failure criterion used is a
tensile strength criterion; therefore, the initiation pressure will
be understood herein as the fluid pressure within the borehole
resulting in the initiation of a tensile crack in a defect free
subsurface material.
Workflow
[0029] This subject disclosure considers the following improvements
to take into account the anisotropic nature of the rocks and is
further depicted in the workflow in FIG. 3. FIG. 3 illustrates an
embodiment of the subject disclosure. FIG. 3 illustrates a workflow
for determining perforation orientations for hydraulic fracturing
in anisotropic earth formations. The workflow comprises a plurality
of steps as illustrated in FIG. 3.
[0030] The first step is determination of anisotropic rock
properties (301). This step involves (1) the acquisition of
wireline or Logging While Drilling (LWD) sonic logs with all modes
(monopole, dipole and Stoneley) with a 3D deviation survey, and (2)
data processing to identify and estimate borehole sonic anisotropy.
This step is performed using tools and procedures which have been
described. See U.S. Pat. No. 6,714,480 to Sinha et al, entitled
"Determination of anisotropic moduli of earth formations", U.S.
Pat. No. 6,718,266 to Sinha et al., entitled "Determination of
dipole shear anisotropy of earth formations", U.S. Patent
Publication No.: 2009-0210160 to Suarez-Rivera et al. entitled
"Estimating horizontal stress from three-dimensional anisotropy"
and U.S. Pat. No. 8,117,014 to Prioul et al, entitled "Methods to
estimate subsurface deviatoric stress characteristics from borehole
sonic log anisotropy directions and image log failure directions".
For TTI media, this leads to five elastic constants, e.g. c.sub.11,
c.sub.33, c.sub.13, c.sub.44 and c.sub.66, and two angles (the dip
azimuth (.beta..sub.A and dip angle .beta..sub.D of the TI plane,
as described above). The five elastic constants will define the
stiffness tensor in the TI frame which can be inverted to get the
compliance tensor rotated in the borehole frame and noted a.sub.ij.
This step can be completed for wells of arbitrary orientation.
[0031] The second step is the determination of far-field stresses
in anisotropic formations (303). In embodiments of the subject
disclosure it is assumed that the principal stress field
(.sigma.1>.sigma.2>.sigma.3) and the pore pressure P.sub.p
are given but important considerations to estimate far-field
stresses in anisotropic formations are considered. See United
States Patent Publication No.: 2009-0210160 to Suarez-Rivera et al.
entitled "Estimating horizontal stress from three-dimensional
anisotropy". For example, this includes taking into account the
anisotropy of the rock in the determination of the gravitational
component of the stress field which leads to a relationship between
the vertical and horizontal stresses for a transversely isotropic
rocks with vertical axis of symmetry (VTI) or a titled axis of
symmetry (TTI, see FIG. 2 above) such as described respectively by
Thiercelin and Plumb (1994, SPE 21847), Amadei & Pan (1992,
IJRMMS) and United States Patent Publication No.: 2009-0210160 to
Suarez-Rivera et al. entitled "Estimating horizontal stress from
three-dimensional anisotropy". The stress tensor is rotated in the
TOH frame, to get .sigma..sub.TOH=[.sigma..sub.xx.sup.TOH
.sigma..sub.xy.sup.TOH .sigma..sub.xz.sup.TOH;
.sigma..sub.xy.sup.TOH .sigma..sub.yy.sup.TOH
.sigma..sub.yz.sup.TOH; .sigma..sub.xz.sup.TOH
.sigma..sub.yz.sup.TOH .sigma..sub.zz.sup.TOH].
[0032] The third step is the determination of borehole stresses in
an anisotropic formation (305). In embodiments of the subject
disclosure a general solution for the stresses around a borehole in
an anisotropic medium can be found using elasticity results from
the superposition of the far field in-situ stress tensor
.sigma..sub.TOH and the general expressions for the
borehole-induced stresses (.sigma..sub.bi). See B. Amadei, Rock
Anisotropy and the theory of stress measurements. Lecture notes in
engineering. Springer Verlag, 1983, S. G. Lekhnitskii, Theory of
elasticity of an anisotropic body. MIR Publishers, Moscow, 1963,
Gaede, O., Karpfinger, F., Jocker, J. & Prioul, R., Comparison
between analytical and 3D finite element solutions for borehole
stresses in anisotropic elastic rock, International Journal of Rock
Mechanics & Mining Sciences, 2012, 51, 53-63. This step applies
to arbitrary well orientation, arbitrary stress field and arbitrary
elastic anisotropy of the formation. [0033] The stress components
in the plane orthogonal to the borehole are in Cartesian
coordinates:
[0033]
.sigma..sub.xx,BH=.sigma..sub.xx,TOH+.sigma..sub.xx,bi=.sigma..su-
b.xx,TOH+2Re[.mu..sub.1.sup.2.phi.'.sub.1(z.sub.1)+.mu..sub.2.sup.2.phi.'.-
sub.2(z.sub.2)+.lamda..sub.3.mu..sub.3.sup.2.phi.'.sub.3(z.sub.3)]
.sigma..sub.yy,BH=.sigma..sub.yy,TOH+.sigma..sub.yy,bi=.sigma..sub.yy,TO-
H+2Re[.phi.'.sub.1(z.sub.1)+.phi.'.sub.2(z.sub.2)+.lamda..sub.3.phi.'.sub.-
3(z.sub.3)]
.sigma..sub.xy,BH=.sigma..sub.xy,TOH+.sigma..sub.xy,bi=.sigma..sub.xy,TO-
H+2Re[.mu..sub.1.phi.'.sub.1(z.sub.1)+.mu..sub.2.phi.'.sub.2(z.sub.2)+.lam-
da..sub.3.mu..sub.3.phi.'.sub.3(z.sub.3)]
.sigma..sub.xz,BH=.sigma..sub.xz,TOH+.sigma..sub.xz,bi=.sigma..sub.xz,TO-
H+2Re[.lamda..sub.1.mu..sub.1.phi.'.sub.1(z.sub.1)+.lamda..sub.2.mu..sub.2-
.phi.'.sub.2(z.sub.2)+.lamda..sub.3.mu..sub.3.phi.'.sub.3(z.sub.3)]
.sigma..sub.yz,BH=.sigma..sub.yz,TOH+.sigma..sub.yz,bi=.sigma..sub.yz,TO-
H+2Re[.lamda..sub.1.phi.'.sub.1(z.sub.1)+.lamda..sub.2.phi.'.sub.2(z.sub.2-
)+.phi.'.sub.3(z.sub.3)] [0034] The stress component in the
borehole axis direction is deduced from the generalized plane
stress assumption using the other stress components and the
compliance tensor component a.sub.ij:
[0034] .sigma. zz , BH = .sigma. zz , TOH - 1 a 33 ( a 31 .sigma.
xx , bi + a 32 .sigma. yy , bi + a 34 .sigma. yz , bi + a 35
.sigma. xz , bi + a 36 .sigma. xy , bi ) ##EQU00003## [0035] The
Cartesian stresses are then transformed into cylindrical
coordinates to get .sigma..sub.rr, .sigma..sub..theta..theta.,
.sigma..sub.zz, .sigma..sub..theta.z, .sigma..sub.r.theta.,
.sigma..sub.rz [0036] These equations include the solutions to
compute borehole stresses away from the borehole wall at a desired
radial position into the formation. The fourth step is the
determination of an optimum perforation orientation (307). This
step is the same as for an isotropic rock. The ideal perforation
orientation for tensile initiation is found for the azimuthal
position .theta..sub.t and the initiation pressure P.sub.w.sup.init
where the minimum principal stress at the borehole wall is given
by
[0036] .sigma. t = .sigma. zz + .sigma. .theta..theta. 2 - (
.sigma. zz - .sigma. .theta..theta. 2 ) 2 + .sigma. .theta. z 2 = -
To + Pp , ##EQU00004##
where To is the tensile strength of the rock and Pp is the pore
pressure. Steps three and four can be performed not only at the
borehole wall but at any desired radial position within the
formation using the appropriate stress concentration solutions from
step 3.
[0037] The fifth step is to perforate a well in an optimum
orientation (309). Knowing the optimum orientation, a perforation
tool may be lowered into a well, the tool perforating the well in
the direction of the optimum orientation obtained from the previous
step.
[0038] In addition to the previous steps at a given depth point, it
is understood that since anisotropic rock properties and far-field
stress properties (from steps 1 and 2 above) can vary along the
well, borehole stresses (step 3 above) will vary along the borehole
and therefore step 3 can be used to select the depth points with
similar near-wellbore stresses or similar wellbore fluid initiation
pressure where to place perforation clusters for a given hydraulic
fracturing stage in rock. Therefore, step 3 may be used with the
borehole stresses to determine how to place hydraulic fracturing
stages along the well.
Example
[0039] If the following conditions are considered at a given depth
for a hypothetical well: [0040] The stress field is the result of
step 1: .sigma..sub.v=19.98 MPa, .sigma..sub.H=19.9 MPa,
.sigma..sub.h=18.73 MPa, P.sub.p=11.63 MPa. .sigma..sub.H is
oriented in the North direction. [0041] The anisotropic material
properties are the result of step 2: E.sub.h=3.55 GPa, E.sub.v=2.13
GPa, .upsilon..sub.h=0.4, .upsilon..sub.v=0.29, G.sub.h=1.27 GPa.
The dip azimuth and dip angle are both zero here
(.beta..sub.D=.beta..sub.A=0).
[0042] If we loop over a grid of well orientation with deviation
angle between 0 and 90.degree. and azimuth between 0 and
360.degree., we can perform steps 3 and 4 for each well orientation
to get the ideal azimuthal position .theta..sup.TTI.sub.t and the
wellbore fluid initiation pressure P.sub.w.sup.init.sub.TTI. If
step 3 of this workflow is replaced by its isotropic version
(described in the background) using the horizontal Poisson's ratio
as a material property, we can compute azimuthal position
.theta..sup.ISO.sub.t and the wellbore fluid initiation pressure
P.sub.w.sup.init.sub.ISO and compare the difference between those
two angles. Results on FIG. 4A-4C show that the difference
|.theta..sup.TTI.sub.t-.theta..sup.ISO.sub.t| due to the anisotropy
of the material orientation which can be up to 45.degree. in this
example (Difference between P.sub.w.sup.init.sub.TTI and
P.sub.w.sup.init.sub.ISO not shown here, for such details, we refer
to Prioul, R., Karpfinger, F., Deenadayalu, C. & Suarez-Rivera,
R, Improving Fracture Initiation Predictions on Arbitrarily
Oriented Wells in Anisotropic Shales, Society of Petroleum
Engineers, SPE-147462, 2011, 1, 1-18).
[0043] FIG. 4A-C depicts examples of an optimum perforation
orientation angle around a borehole computed using in FIG. 4A
isotropic stress concentration and in FIG. 4B anisotropic stress
concentrations. The difference between FIG. 4A and FIG. 4B is shown
in FIG. 4C. Results are plotted on a polar grid where each point of
the grid correspond to well orientation, with radial variation
corresponding to well deviation (from 0 to 90) and azimuthal
variation corresponding to well azimuth (from 0 to 360) with the
convention of clockwise positive rotation from North to East.
[0044] Although only a few example embodiments have been described
in detail above, those skilled in the art will readily appreciate
that many modifications are possible in the example embodiments
without materially departing from this invention. Accordingly, all
such modifications are intended to be included within the scope of
this disclosure as defined in the following claims. In the claims,
means-plus-function clauses are intended to cover the structures
described herein as performing the recited function and not only
structural equivalents, but also equivalent structures. Thus,
although a nail and a screw may not be structural equivalents in
that a nail employs a cylindrical surface to secure wooden parts
together, whereas a screw employs a helical surface, in the
environment of fastening wooden parts, a nail and a screw may be
equivalent structures. It is the express intention of the applicant
not to invoke 35 U.S.C. .sctn.112, paragraph 6 for any limitations
of any of the claims herein, except for those in which the claim
expressly uses the words `means for` together with an associated
function.
* * * * *